Solving Kepler's Problem: Understanding Awkward Integration

[Piano man]

Hi,
I'm reading up on Kepler's problem at the moment, and there's a step in the book that I don't understand.

Starting off with the equation of the path $$\phi=\int\frac{M dr/r^2}{\sqrt{2m[E-U(r)]-M^2/r^2}}+\mbox{constant}$$

The step involves subbing in $$U=-\alpha/r$$, 'and effecting elementary integration' to get
$$\phi=\cos^{-1}\frac{(M/r)-(m\alpha/M)}{\sqrt{(2mE+\frac{m^2\alpha^2}{M^2}})}+\mbox{constant}$$

But it doesn't look very elementary to me

Does anyone have any idea?

Thanks.

 

[vela]

Try the substitution w=1/r and then complete the square in the radical.